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Power series for the secans

The secans can be expressed as the power series

 


Explanation

The first derivative of the secans is

Only the secans and the tangent are used. If we determine further derivatives this will remain the same, because (tan x)' = sec2x, so







The point x = 0 gives sec (0) = 1 and tan (0) = 0 so









We substitute this in the Taylor series and find

 


General format

You can write the power series as a sum

where Un is the n-th up/down number. Or

where E2n is the n-th Euler number.

 


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